Does there exist $u\in H^2(\Bbb{R}^n)$ so that $-\Delta u=\lambda u$.
Of course if such $u$ exists, it must be a smooth function by elliptic estimate and bootstrap strategy. So we need to find $u$ so that $-\Delta u=\lambda u$ in classical sense with norm constrain.
I don't think such an eigenfunction exist, but I can't prove it.
Thanks for advance for anyone who could offer help.
